Determine if Odd, Even, or Neither x^4-2x^2+9

$x^{4} - 2 x^{2} + 9$

Write $x^{4} - 2 x^{2} + 9$ as a function.

$f (x) = x^{4} - 2 x^{2} + 9$

Find $f (- x)$ .

Find $f (- x)$ by substituting $- x$ for all occurrence of $x$ in $f (x)$ .

$f (- x) = {(- x)}^{4} - 2 {(- x)}^{2} + 9$

Simplify each term.

Apply the product rule to $- x$ .

$f (- x) = {(- 1)}^{4} x^{4} - 2 {(- x)}^{2} + 9$

Raise $- 1$ to the power of $4$ .

$f (- x) = 1 x^{4} - 2 {(- x)}^{2} + 9$

Multiply $x^{4}$ by $1$ .

$f (- x) = x^{4} - 2 {(- x)}^{2} + 9$

Apply the product rule to $- x$ .

$f (- x) = x^{4} - 2 ({(- 1)}^{2} x^{2}) + 9$

Raise $- 1$ to the power of $2$ .

$f (- x) = x^{4} - 2 (1 x^{2}) + 9$

Multiply $x^{2}$ by $1$ .

$f (- x) = x^{4} - 2 x^{2} + 9$

A function is even if $f (- x) = f (x)$ .

Check if $f (- x) = f (x)$ .

$x^{4} - 2 x^{2} + 9 = x^{4} - 2 x^{2} + 9$

Since $x^{4} - 2 x^{2} + 9 = x^{4} - 2 x^{2} + 9$ , the function is even.

The function is even